# A CHARACTERIZATION OF WEIGHTED BERGMAN-PRIVALOV SPACES ON THE UNIT BALL OF Cn

• Matsugu, Yasuo (Department of Mathematical Sciences Faculty of Science Shinshu University) ;
• Miyazawa, Jun (Department of Mathematical Sciences Faculty of Science Shinshu University) ;
• Ueki, Sei-Ichiro (Department of Mathematical Sciences Faculty of Science Shinshu University)
• Published : 2002.09.01
• 97 17

#### Abstract

Let B denote the unit ball in $C^n$, and ν the normalized Lebesgue measure on B. For $\alpha$ > -1, define $dv_\alpha$(z) ＝ $c_\alpha$$(1-\midz\mid^2)^{\alpha}dν(z), z \in B. Here c_\alpha is a positive constant such that v_\alpha(B) ＝ 1. Let H(B) denote the space of all holomorphic functions in B. For p\geq1, define the Bergman-Privalov space (AN)^{p}(v_\alpha) by (AN)^{p}(v_\alpha) = {f\inH(B) : \int_B{log(1+\midf\mid)}^pdv_\alpha\;<\;\infty} In this paper we prove that a function f\inH(B) is in (AN)^{p}$$(v_\alpha)$ if and only if $(1+\midf\mid)^{-2}{log(1+\midf\mid)}^{p-2}\mid\nablaf\mid^2\;\epsilon\;L^1(v_\alpha)$ in the case 1＜p＜$\infty$, or $(1+\midf\mid)^{-2}\midf\mid^{-1}\mid{\nabla}f\mid^2\;\epsilon\;L^1(v_\alpha)$ in the case p ＝ 1, where $nabla$f is the gradient of f with respect to the Bergman metric on B. This is an analogous result to the characterization of the Hardy spaces by M. Stoll [18] and that of the Bergman spaces by C. Ouyang-W. Yang-R. Zhao [13].

#### Keywords

Bergman-Privalov spaces;Privalov spaces;Bergman spaces;Riesz measure;Hardy-Orlicz spaces

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#### Cited by

1. Characterizations of Hardy-Orlicz and Bergman-Orlicz spaces vol.141, pp.5, 2007, https://doi.org/10.1007/s10958-007-0059-8